Question

Let \(F\) be a field and \(f / g \in F(x)\) a rational function, with \(f, g \in F[x]\) such that \(g \neq 0\) is monic and \(\operatorname{gcd}(f, g)=1\). We say that \(f / g\) is defined at a point \(u \in F\) if \(g(u) \neq 0\), and then the value of the rational function at \(u\) is \(f(u) / g(u)\). Let \(f^{*} / g^{*} \in F(x)\) be another rational function, with \(g^{*} \neq 0\) monic and \(\operatorname{gcd}\left(f^{*}, g^{*}\right)=1\), such that \(f / g\) and \(f^{*} / g^{*}\) are defined and their values coincide at \(n=\max \left\\{\operatorname{deg} f+\operatorname{deg} g^{*}, \operatorname{deg} f^{*}+\operatorname{deg} g\right\\}+1\) distinct points \(u_{0}, \ldots, u_{n-1} \in F\). Prove that \(f=f^{*}\) and \(g=g^{*}\).

Short answer

Since \( f g^* = f^* g \) at more points than the sum of their degrees, it follows that \( f = f^* \) and \( g = g^* \). This is derived from polynomial interpolation principles and the given gcd condition.

Step-by-step solution

Understand the Given Conditions

We are given two rational functions \( \frac{f}{g} \) and \( \frac{f^*}{g^*} \) where \( f, g, f^*, g^* \in F[x] \), and both \( g \) and \( g^* \) are monic. Additionally, \( \operatorname{gcd}(f, g) = 1 \) and \( \operatorname{gcd}(f^*, g^*) = 1 \). Furthermore, these functions are equal at \( n = \max\{\operatorname{deg} f + \operatorname{deg} g^*, \operatorname{deg} f^* + \operatorname{deg} g\} + 1 \) distinct points. Our goal is to prove that this implies \( f = f^* \) and \( g = g^* \).

Recognize the Degrees of the Equations

Since \( \frac{f}{g} = \frac{f^*}{g^*} \) at \( n \) distinct points, we need to use the information about the degrees involved: \( \operatorname{deg}(f) + \operatorname{deg}(g^*) \) and \( \operatorname{deg}(f^*) + \operatorname{deg}(g) \). The number of points at which the functions are equal is one more than the maximum of these sums. This information is crucial for showing uniqueness of the expression as a rational function.

Use Polynomial Interpolation Theorem

Polynomials of degree at most \( d \) that agree at \( d + 1 \) distinct points are identical. The given functions \( \frac{f}{g} \) and \( \frac{f^*}{g^*} \) are rational functions, meaning both the numerator and the denominator are polynomials. Since their values coincide at more points than the sum of the degrees, we can apply the polynomial interpolation concept.

Set Up Equation for Equality

Considering that \( \frac{f}{g} = \frac{f^*}{g^*} \) means \( f g^* = f^* g \) for these \( n \) distinct points when simplified. This equation implies that \( f g^* - f^* g = 0 \) is a polynomial that can only be zero if \( f g^* \) and \( f^* g \) are equal polynomials. Given our degree constraints, any non-zero polynomial must have at most \( n-1 \) roots, but the condition here is \( n \). Thus, the polynomial must be zero everywhere.

Infer the Result from the Equation

Since \( f g^* = f^* g \) as polynomials and considering they are defined at more points than the sum of their degrees, with the polynomials fulfilling \( \operatorname{gcd} conditions \), the only solution is that \( f = f^* \) and \( g = g^* \). This follows because any true solution to a polynomial equal to zero, given coprime constraints and degree considerations, implies component equality.

Key concepts

Rational Functions

Rational functions are expressions that represent the ratio of two polynomials. They are written in the form \( \frac{f(x)}{g(x)} \), where both \( f(x) \) and \( g(x) \) are polynomials, and \( g(x) eq 0 \).
Rational functions are an essential part of algebra because they allow us to understand more complex behaviors in equations.
Key things to remember about rational functions:

• They can have undefined points where the denominator is zero.
• The degree of a rational function is determined by the degrees of its numerator and denominator.
• They can display vertical asymptotes, horizontal asymptotes, or oblique asymptotes based on the degree comparison of \( f(x) \) and \( g(x) \).

In this exercise, our rational functions \( \frac{f}{g} \) and \( \frac{f^*}{g^*} \) are from the field \( F(x) \), and they are defined at points where the denominator is not zero. This concept is crucial when proving the equality of these rational functions.

Field Theory

Field theory is a branch of algebra that studies the properties of fields. A field is a set equipped with two operations: addition and multiplication, which satisfy specific properties similar to those in the real numbers. Fields are pivotal in understanding the structure and function of rational functions.
In the context of this problem, knowing about fields helps us appreciate the rational functions' behavior.

• Fields do not allow division by zero, ensuring that whenever we work with rational functions, the denominators are never zero, except at undefined points.
• Polynomials over fields maintain integrity and play a crucial role in various algebraic operations.

Understanding field theory aids in ensuring that there is a foundational integrity to the work done with rational functions, as seen when \( f/g \) must be in reduced form, meaning \( \operatorname{gcd}(f, g) = 1 \).

Interpolation Theorem

The Interpolation Theorem provides tools for reconstructing polynomials that are specified by their values at given distinct points. A significant result is that no two distinct polynomials can coincide at more than a certain number of points without being identical.
This theorem is especially useful when considering the equality of rational functions over a field.

• For polynomials of degree \( d \), if they agree at \( d + 1 \) distinct points, they must be identical.
• Using this idea, rational functions, which are formed of numerators and denominators that share certain values at given points, can be examined for equality.

In the exercise, we prove that \( \frac{f}{g} = \frac{f^*}{g^*} \) by showing the rational components agree at enough distinct points to necessitate the equality of numerators and denominators separately.

Polynomial Degree

The degree of a polynomial is critical in understanding its behavior and properties. The degree is the highest power of the variable in the polynomial and dictates many features, such as:

• The number of roots a polynomial can have.
• The end behavior as the variable tends towards infinity or negative infinity.
• The shape and complexity of the graph of the polynomial.

In rational functions, the combined degrees of the numerator and the denominator are instrumental in defining the function's maximum number of distinct points of coincidence.
This idea is applied in the exercise by examining where the rational functions \( \frac{f}{g} \) and \( \frac{f^*}{g^*} \) share these points. By understanding the relationship between degrees and points of agreement, we conclude that enough points beyond the degree reach must mean both functions express identical polynomials.